For reference this is from page 27 here
Consider the stochastic heat equation$$ \begin{cases} \dfrac{\partial u}{\partial t}(x,t)=\dfrac{\partial ^2 u}{\partial x^2}(x,t) +f(u(x,t))\dot{W}(x,t) & t >0, x\in[0,L] \\ \dfrac{\partial u}{\partial x}(0,t)=\dfrac{\partial u}{\partial x}(L,t)=0 &t>0 \\ u(x,0)=u_0(x) &x \in[0,L] \end{cases} $$$\dot{W}(x,s$) is refered to as the space-time white noise. But this doesn't make sense to the defintion of white noise given in the same text.
$\textbf{White Noise}$ here is defined as follows: The gaussian process $\{\dot{W}(A)\}_{A \in \mathscr B(\mathbb R^n)}$ with $E(\dot{W}(A))=0$ and $E(\dot{W}(A)\dot{W}(B))=\lambda^n(A\cap B)$ where $\lambda ^n$ is the n-dimensianal Lesbegue measure is called White Noise.
It then goes on to say that we can consider for $A \in \mathscr B(\mathbb R^n)$ that $\dot{W}(A) = \int 1_A dW$ and for $h\in L^2(\mathbb R^n)$ we define $\dot{W}(h) = \int h(t)dW(t)$ which is the standard Wiener Integral.
This is also the same definition given in the Walsh notes
So my question is, what is $\dot{W}(x,t)$? Could it be $\dot{W}(0\times t) \times (0,x])$?
Also how does this definition relate to the definitions given in this question? In particular to notion that White noise, $\dot{W}$, is the weak derivitive of Brownian motion. Ie: If $W_t$ is a standard Brownian motion then $\dot{W}$ is a distribution on the set of test functions such that $$ -\int W_t\dfrac{\partial \phi}{\partial t}(t)dt=\int\dot{W}_t\phi(t)dt $$
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